Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in $ {\bf R}\sp 3$

Author: V. Girault
Journal: Math. Comp. 51 (1988), 55-74
MSC: Primary 65N30; Secondary 76-08, 76D05
MathSciNet review: 942143
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the steady state, incompressible Navier-Stokes equations with nonstandard boundary conditions of the form $ {\mathbf{u}} \cdot {\mathbf{n}} = 0$, $ \mathbf{curl}\;{\mathbf{u}} \times {\mathbf{n}} = {\mathbf{0}}$, either on the entire boundary or mixed with the standard boundary condition $ {\mathbf{u}} = {\mathbf{0}}$ on part of the boundary. The problem is expressed in terms of vector potential, vorticity and pressure. The vorticity and vector potential are approximated with curl-conforming finite elements and the pressure with standard continuous finite elements. The error estimates yield nearly optimal results for the purely nonstandard problem.

References [Enhancements On Off] (What's this?)

  • [1] Jean-Pierre Aubin, Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Gelerkin’s and finite difference methods, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 599–637. MR 0233068
  • [2] Catherine Bègue, Carlos Conca, François Murat, and Olivier Pironneau, À nouveau sur les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 1, 23–28 (French, with English summary). MR 878818
  • [3] C. Bègue, C. Conca, F. Murat & O. Pironneau, "Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression," Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar (H. Brézis and J. L. Lions, eds.) (To appear.)
  • [4] A. Bendali, J. M. Domínguez, and S. Gallic, A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains, J. Math. Anal. Appl. 107 (1985), no. 2, 537–560. MR 787732,
  • [5] C. Bernardi, Méthode d'Éléments Finis Mixtes pour les Équations de Navier-Stokes, Thèse, Univ. Paris VI, 1979.
  • [6] Christine Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal. 26 (1989), no. 5, 1212–1240 (English, with French summary). MR 1014883,
  • [7] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [8] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
  • [9] M. Dauge, Personal communication.
  • [10] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches Mathématiques, No. 21. MR 0464857
  • [11] V. Girault, "Elementos finitos mixtos para ecuaciones de Navier-Stokes en $ {{\mathbf{R}}^3}$," Acta Ciént. Venezolana. (To appear).
  • [12] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • [13] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
  • [14] Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR 0466912
  • [15] J. C. Nédélec, "Mixed finite elements in $ {{\mathbf{R}}^3}$," Numer. Math., v. 35, 1980, pp. 315-341.
  • [16] J.-C. Nédélec, Éléments finis mixtes incompressibles pour l’équation de Stokes dans 𝑅³, Numer. Math. 39 (1982), no. 1, 97–112 (French, with English summary). MR 664539,
  • [17] J.-C. Nédélec, A new family of mixed finite elements in 𝑅³, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305,
  • [18] J. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math. 11 (1968), 346–348 (German). MR 0233502,
  • [19] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
  • [20] Reinhard Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Numér. 12 (1978), no. 1, 85–90, iii (English, with French summary). MR 0483557
  • [21] R. Témam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.
  • [22] Rüdiger Verfürth, Mixed finite element approximation of the vector potential, Numer. Math. 50 (1987), no. 6, 685–695. MR 884295,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 76-08, 76D05

Retrieve articles in all journals with MSC: 65N30, 76-08, 76D05

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society